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<div class="header">
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<div class="title">Householder.h</div>  </div>
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<div class="contents">
<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">// This file is part of Eigen, a lightweight C++ template library</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment">// for linear algebra.</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment">// Copyright (C) 2010 Benoit Jacob &lt;jacob.benoit.1@gmail.com&gt;</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment">// Copyright (C) 2009 Gael Guennebaud &lt;gael.guennebaud@inria.fr&gt;</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment">// This Source Code Form is subject to the terms of the Mozilla</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment">// Public License v. 2.0. If a copy of the MPL was not distributed</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment">// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160; </div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="preprocessor">#ifndef EIGEN_HOUSEHOLDER_H</span></div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="preprocessor">#define EIGEN_HOUSEHOLDER_H</span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160; </div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;<span class="preprocessor">#include &quot;./InternalHeaderCheck.h&quot;</span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160; </div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;<span class="keyword">namespace </span><a class="code" href="namespaceEigen.html">Eigen</a> { </div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160; </div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;<span class="keyword">namespace </span>internal {</div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;<span class="keyword">template</span>&lt;<span class="keywordtype">int</span> n&gt; <span class="keyword">struct </span>decrement_size</div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;{</div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;  <span class="keyword">enum</span> {</div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;    ret = n==<a class="code" href="namespaceEigen.html#ad81fa7195215a0ce30017dfac309f0b2">Dynamic</a> ? n : n-1</div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;  };</div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;};</div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;}</div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160; </div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Derived&gt;</div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l00045"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixBase.html#aebf4bac7dffe2685ab93734fb776e817">   45</a></span>&#160;<span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixBase.html#aebf4bac7dffe2685ab93734fb776e817">MatrixBase&lt;Derived&gt;::makeHouseholderInPlace</a>(<a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a>&amp; tau, RealScalar&amp; beta)</div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;{</div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;  <a class="code" href="classEigen_1_1VectorBlock.html">VectorBlock&lt;Derived, internal::decrement_size&lt;Base::SizeAtCompileTime&gt;::ret</a>&gt; essentialPart(derived(), 1, size()-1);</div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;  makeHouseholder(essentialPart, tau, beta);</div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;}</div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160; </div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Derived&gt;</div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> EssentialPart&gt;</div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l00069"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixBase.html#a13291e900f7e81ddc6e5b8802f82092b">   69</a></span>&#160;<span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixBase.html#a13291e900f7e81ddc6e5b8802f82092b">MatrixBase&lt;Derived&gt;::makeHouseholder</a>(</div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;  EssentialPart&amp; essential,</div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;  <a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a>&amp; tau,</div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;  RealScalar&amp; beta)<span class="keyword"> const</span></div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;<span class="keyword"></span>{</div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;  <span class="keyword">using</span> std::sqrt;</div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;  <span class="keyword">using</span> numext::conj;</div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;  </div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;  EIGEN_STATIC_ASSERT_VECTOR_ONLY(EssentialPart)</div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;  <a class="code" href="classEigen_1_1VectorBlock.html">VectorBlock&lt;const Derived, EssentialPart::SizeAtCompileTime&gt;</a> tail(derived(), 1, size()-1);</div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;  </div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;  RealScalar tailSqNorm = size()==1 ? RealScalar(0) : tail.squaredNorm();</div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;  <a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a> c0 = coeff(0);</div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;  <span class="keyword">const</span> RealScalar tol = (std::numeric_limits&lt;RealScalar&gt;::min)();</div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160; </div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;  <span class="keywordflow">if</span>(tailSqNorm &lt;= tol &amp;&amp; numext::abs2(numext::imag(c0))&lt;=tol)</div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;  {</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;    tau = RealScalar(0);</div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;    beta = numext::real(c0);</div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;    essential.setZero();</div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;  }</div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;  <span class="keywordflow">else</span></div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;  {</div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;    beta = <a class="code" href="namespaceEigen.html#af4f536e8ea56702e63088efb3706d1f0">sqrt</a>(numext::abs2(c0) + tailSqNorm);</div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;    <span class="keywordflow">if</span> (numext::real(c0)&gt;=RealScalar(0))</div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;      beta = -beta;</div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;    essential = tail / (c0 - beta);</div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;    tau = <a class="code" href="namespaceEigen.html#ab84f39a06a18e1ebb23f8be80345b79d">conj</a>((beta - c0) / beta);</div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;  }</div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;}</div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160; </div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Derived&gt;</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> EssentialPart&gt;</div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l00118"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixBase.html#a8f2c8059ef3f04cfa0c73b4c012db855">  118</a></span>&#160;<span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixBase.html#a8f2c8059ef3f04cfa0c73b4c012db855">MatrixBase&lt;Derived&gt;::applyHouseholderOnTheLeft</a>(</div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;  <span class="keyword">const</span> EssentialPart&amp; essential,</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;  <span class="keyword">const</span> <a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a>&amp; tau,</div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;  <a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a>* workspace)</div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;{</div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;  <span class="keywordflow">if</span>(rows() == 1)</div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;  {</div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;    *<span class="keyword">this</span> *= <a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a>(1)-tau;</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;  }</div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;  <span class="keywordflow">else</span> <span class="keywordflow">if</span>(!numext::is_exactly_zero(tau))</div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;  {</div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;    <a class="code" href="classEigen_1_1Map.html">Map&lt;typename internal::plain_row_type&lt;PlainObject&gt;::type</a>&gt; tmp(workspace,cols());</div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;    <a class="code" href="classEigen_1_1Block.html">Block&lt;Derived, EssentialPart::SizeAtCompileTime, Derived::ColsAtCompileTime&gt;</a> bottom(derived(), 1, 0, rows()-1, cols());</div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;    tmp.noalias() = essential.adjoint() * bottom;</div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;    tmp += this-&gt;row(0);</div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;    this-&gt;row(0) -= tau * tmp;</div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;    bottom.noalias() -= tau * essential * tmp;</div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;  }</div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;}</div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160; </div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> Derived&gt;</div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;<span class="keyword">template</span>&lt;<span class="keyword">typename</span> EssentialPart&gt;</div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;EIGEN_DEVICE_FUNC</div>
<div class="line"><a name="l00156"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixBase.html#ab3e52262b41fa40e194dda245e0f9675">  156</a></span>&#160;<span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixBase.html#ab3e52262b41fa40e194dda245e0f9675">MatrixBase&lt;Derived&gt;::applyHouseholderOnTheRight</a>(</div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;  <span class="keyword">const</span> EssentialPart&amp; essential,</div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;  <span class="keyword">const</span> <a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a>&amp; tau,</div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;  <a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a>* workspace)</div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;{</div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;  <span class="keywordflow">if</span>(cols() == 1)</div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;  {</div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;    *<span class="keyword">this</span> *= <a class="code" href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Scalar</a>(1)-tau;</div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;  }</div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;  <span class="keywordflow">else</span> <span class="keywordflow">if</span>(!numext::is_exactly_zero(tau))</div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;  {</div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;    <a class="code" href="classEigen_1_1Map.html">Map&lt;typename internal::plain_col_type&lt;PlainObject&gt;::type</a>&gt; tmp(workspace,rows());</div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;    <a class="code" href="classEigen_1_1Block.html">Block&lt;Derived, Derived::RowsAtCompileTime, EssentialPart::SizeAtCompileTime&gt;</a> right(derived(), 0, 1, rows(), cols()-1);</div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;    tmp.noalias() = right * essential;</div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160;    tmp += this-&gt;col(0);</div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;    this-&gt;col(0) -= tau * tmp;</div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;    right.noalias() -= tau * tmp * essential.adjoint();</div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;  }</div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;}</div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160; </div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;} <span class="comment">// end namespace Eigen</span></div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160; </div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;<span class="preprocessor">#endif </span><span class="comment">// EIGEN_HOUSEHOLDER_H</span></div>
<div class="ttc" id="aclassEigen_1_1Block_html"><div class="ttname"><a href="classEigen_1_1Block.html">Eigen::Block</a></div><div class="ttdoc">Expression of a fixed-size or dynamic-size block.</div><div class="ttdef"><b>Definition:</b> Block.h:107</div></div>
<div class="ttc" id="aclassEigen_1_1DenseBase_html_a5feed465b3a8e60c47e73ecce83e39a2"><div class="ttname"><a href="classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Eigen::DenseBase::Scalar</a></div><div class="ttdeci">internal::traits&lt; Derived &gt;::Scalar Scalar</div><div class="ttdef"><b>Definition:</b> DenseBase.h:61</div></div>
<div class="ttc" id="aclassEigen_1_1Map_html"><div class="ttname"><a href="classEigen_1_1Map.html">Eigen::Map</a></div><div class="ttdoc">A matrix or vector expression mapping an existing array of data.</div><div class="ttdef"><b>Definition:</b> Map.h:98</div></div>
<div class="ttc" id="aclassEigen_1_1MatrixBase_html_a13291e900f7e81ddc6e5b8802f82092b"><div class="ttname"><a href="classEigen_1_1MatrixBase.html#a13291e900f7e81ddc6e5b8802f82092b">Eigen::MatrixBase::makeHouseholder</a></div><div class="ttdeci">void makeHouseholder(EssentialPart &amp;essential, Scalar &amp;tau, RealScalar &amp;beta) const</div><div class="ttdef"><b>Definition:</b> Householder.h:69</div></div>
<div class="ttc" id="aclassEigen_1_1MatrixBase_html_a8f2c8059ef3f04cfa0c73b4c012db855"><div class="ttname"><a href="classEigen_1_1MatrixBase.html#a8f2c8059ef3f04cfa0c73b4c012db855">Eigen::MatrixBase::applyHouseholderOnTheLeft</a></div><div class="ttdeci">void applyHouseholderOnTheLeft(const EssentialPart &amp;essential, const Scalar &amp;tau, Scalar *workspace)</div><div class="ttdef"><b>Definition:</b> Householder.h:118</div></div>
<div class="ttc" id="aclassEigen_1_1MatrixBase_html_ab3e52262b41fa40e194dda245e0f9675"><div class="ttname"><a href="classEigen_1_1MatrixBase.html#ab3e52262b41fa40e194dda245e0f9675">Eigen::MatrixBase::applyHouseholderOnTheRight</a></div><div class="ttdeci">void applyHouseholderOnTheRight(const EssentialPart &amp;essential, const Scalar &amp;tau, Scalar *workspace)</div><div class="ttdef"><b>Definition:</b> Householder.h:156</div></div>
<div class="ttc" id="aclassEigen_1_1MatrixBase_html_aebf4bac7dffe2685ab93734fb776e817"><div class="ttname"><a href="classEigen_1_1MatrixBase.html#aebf4bac7dffe2685ab93734fb776e817">Eigen::MatrixBase::makeHouseholderInPlace</a></div><div class="ttdeci">void makeHouseholderInPlace(Scalar &amp;tau, RealScalar &amp;beta)</div><div class="ttdef"><b>Definition:</b> Householder.h:45</div></div>
<div class="ttc" id="aclassEigen_1_1VectorBlock_html"><div class="ttname"><a href="classEigen_1_1VectorBlock.html">Eigen::VectorBlock</a></div><div class="ttdoc">Expression of a fixed-size or dynamic-size sub-vector.</div><div class="ttdef"><b>Definition:</b> VectorBlock.h:62</div></div>
<div class="ttc" id="anamespaceEigen_html"><div class="ttname"><a href="namespaceEigen.html">Eigen</a></div><div class="ttdoc">Namespace containing all symbols from the Eigen library.</div><div class="ttdef"><b>Definition:</b> Core:139</div></div>
<div class="ttc" id="anamespaceEigen_html_ab84f39a06a18e1ebb23f8be80345b79d"><div class="ttname"><a href="namespaceEigen.html#ab84f39a06a18e1ebb23f8be80345b79d">Eigen::conj</a></div><div class="ttdeci">const Eigen::CwiseUnaryOp&lt; Eigen::internal::scalar_conjugate_op&lt; typename Derived::Scalar &gt;, const Derived &gt; conj(const Eigen::ArrayBase&lt; Derived &gt; &amp;x)</div></div>
<div class="ttc" id="anamespaceEigen_html_ad81fa7195215a0ce30017dfac309f0b2"><div class="ttname"><a href="namespaceEigen.html#ad81fa7195215a0ce30017dfac309f0b2">Eigen::Dynamic</a></div><div class="ttdeci">const int Dynamic</div><div class="ttdef"><b>Definition:</b> Constants.h:24</div></div>
<div class="ttc" id="anamespaceEigen_html_af4f536e8ea56702e63088efb3706d1f0"><div class="ttname"><a href="namespaceEigen.html#af4f536e8ea56702e63088efb3706d1f0">Eigen::sqrt</a></div><div class="ttdeci">const Eigen::CwiseUnaryOp&lt; Eigen::internal::scalar_sqrt_op&lt; typename Derived::Scalar &gt;, const Derived &gt; sqrt(const Eigen::ArrayBase&lt; Derived &gt; &amp;x)</div></div>
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